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Microstrip Filters for RF/Microwave Applications. Jia-Sheng Hong, M. J. Lancaster Copyright © 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-38877-7 (Hardback); 0-471-22161-9 (Electronic)
CHAPTER 2
Network Analysis
Filter networks are essential building elements in many areas of RF/microwave en-gineering. Such networks are used to select/reject or separate/combine signals at different frequencies in a host of RF/microwave systems and equipment. Although the physical realization of filters at RF/microwave frequencies may vary, the circuit network topology is common to all.
At microwave frequencies, voltmeters and ammeters for the direct measurement of voltages and currents do not exist. For this reason, voltage and current, as a mea-sure of the level of electrical excitation of a network, do not play a primary role at microwave frequencies. On the other hand, it is useful to be able to describe the op-eration of a microwave network such as a filter in terms of voltages, currents, and impedances in order to make optimum use of low-frequency network concepts.
It is the purpose of this chapter to describe various network concepts and provide equations that are useful for the analysis of filter networks.
2.1 NETWORK VARIABLES
Most RF/microwave filters and filter components can be represented by a two-port network, as shown in Figure 2.1, where V1, V2 and I1, I2 are the voltage and current variables at the ports 1 and 2, respectively, Z01 and Z02 are the terminal impedances, and Es is the source or generator voltage. Note that the voltage and current variables are complex amplitudes when we consider sinusoidal quantities. For example, a si-nusoidal voltage at port 1 is given by
v1(t) = |V1|cos(vt + f) (2.1)
We can then make the following transformations:
v1(t) = |V1|cos(vt + f) = Re(|V1|ej(vt+f)) = Re(V1ejvt) (2.2)
7
8 NETWORK ANAYLSIS
I1
Z01 a1 V1
Es b1
Two-port network
I2
a2 V2
b2
Z02
FIGURE 2.1 Two-port network showing network variables.
where Re denotes “the real part of” the expression that follows it. Therefore, one can identify the complex amplitude V1 defined by
V1 = |V1|ejf (2.3)
Because it is difficult to measure the voltage and current at microwave frequencies, the wave variables a1, b1 and a2, b2 are introduced, with a indicating the incident waves and b the reflected waves. The relationships between the wave variables and the voltage and current variables are defined as
Vn = ÏwZw(an + bn)
n = 1 and 2 (2.4a) In = ÏwZw (an – bn)
or an = }}1ÏwZw + ÏwZw In2
n = 1 and 2 (2.4b) bn = }}1ÏwZw – ÏwZw In2
The above definitions guarantee that the power at port n is
Pn = –Re(Vn·I*) = –(ana* – bnb*) (2.5)
where the asterisk denotes a conjugate quantity. It can be recognized that ana*/2 is the incident wave power and bnb*/2 is the reflected wave power at port n.
2.2 SCATTERING PARAMETERS
The scattering or S parameters of a two-port network are defined in terms of the wave variables as
2.2 SCATTERING PARAMETERS 9
S11 = }} S12 = }}
1 a2=0 2 a1=0 (2.6)
S21 = }} S22 = }}
1 a2=0 2 a1=0
where an = 0 implies a perfect impedance match (no reflection from terminal im-pedance) at port n. These definitions may be written as
3b14= 3S11 S1243a14 (2.7)
where the matrix containing the S parameters is referred to as the scattering matrix or S matrix, which may simply be denoted by [S].
The parameters S11 and S22 are also called the reflection coefficients, whereas S12 and S21 the transmission coefficients. These are the parameters directly mea-surable at microwave frequencies. The S parameters are in general complex, and it is convenient to express them in terms of amplitudes and phases, i.e., Smn = |Smn|ejfmn for m, n = 1, 2. Often their amplitudes are given in decibels (dB), which are defined as
20 log|Smn| dB m, n = 1, 2 (2.8)
where the logarithm operation is base 10. This will be assumed through this book unless otherwise stated. For filter characterization, we may define two parameters:
LA = –20 log|Smn| dB
LR = 20 log|Snn| dB
m, n = 1, 2(m Þ n)
(2.9) n = 1, 2
where LA denotes the insertion loss between ports n and m and LR represents the re-turn loss at port n. Instead of using the return loss, the voltage standing wave ratio VSWR may be used. The definition of VSWR is
1 + |Snn|
1 – |Snn|
(2.10)
Whenever a signal is transmitted through a frequency-selective network such as a filter, some delay is introduced into the output signal in relation to the input signal. There are other two parameters that play role in characterizing filter performance related to this delay.The first one is the phase delay, defined by
tp = f21 seconds (2.11)
10 NETWORK ANAYLSIS
where f21 is in radians and v is in radians per second. Port 1 is the input port and port 2 is the output port. The phase delay is actually the time delay for a steady sinusoidal signal and is not necessarily the true signal delay because a steady si-nusoidal signal does not carry information; sometimes, it is also referred to as the carrier delay [5]. The more important parameter is the group delay, defined by
td = – df21 seconds (2.12)
This represents the true signal (baseband signal) delay, and is also referred to as the envelope delay.
In network analysis or synthesis, it may be desirable to express the reflection pa-rameter S11 in terms of the terminal impedance Z01 and the so-called input imped-ance Zin1 = V1/I1, which is the impedance looking into port 1 of the network. Such an expression can be deduced by evaluating S11 in (2.6) in terms of the voltage and current variables using the relationships defined in (2.4b). This gives
b1 V1/ÏwZw – ÏZwwI1 11 a1 a2=0 V1/ÏZww + ÏwZwI1
(2.13)
Replacing V1 by Zin1I1 results in the desired expression
Zin1 – Z01 11 Zin1 + Z01
(2.14)
Similarly, we can have
Zin2 – Z02 22 Zin2 + Z02
(2.15)
where Zin2 = V2/I2 is the input impedance looking into port 2 of the network. Equa-tions (2.14) and (2.15) indicate the impedance matching of the network with respect to its terminal impedances.
The S parameters have several properties that are useful for network analysis. For a reciprocal network S12 = S21. If the network is symmetrical, an additional property, S11 = S22, holds. Hence, the symmetrical network is also reciprocal. For a lossless passive network the transmitting power and the reflected power must equal to the to-tal incident power. The mathematical statements of this power conservation condi-tion are
S21S* + S11S* = 1 or |S21|2 + |S11|2 = 1
(2.16)
S12S* + S22S*2 = 1 or |S12|2 + |S22|2 = 1
2.4 OPEN-CIRCUIT IMPEDANCE PARAMETERS 11
2.3 SHORT-CIRCUIT ADMITTANCE PARAMETERS
The short-circuit admittance or Y parameters of a two-port network are defined as
Y11 = }} Y12 = }}
1 V2=0 2 V1=0 (2.17)
Y21 = }} Y22 = }}
1 V2=0 2 V1=0
in which Vn = 0 implies a perfect short-circuit at port n. The definitions of the Y pa-rameters may also be written as
3I14= 3Y11 Y1243V14 (2.18)
where the matrix containing the Y parameters is called the short-circuit admittance or simply Y matrix, and may be denoted by [Y]. For reciprocal networks Y12 = Y21. In addition to this, if networks are symmetrical, Y11 = Y22. For a lossless network, the Y parameters are all purely imaginary.
2.4 OPEN-CIRCUIT IMPEDANCE PARAMETERS
The open-circuit impedance or Z parameters of a two-port network are defined as
Z11 = }} Z12 = }}
1 I2=0 2 I1=0 (2.19)
Z21 = }} Z22 = }}
1 I2=0 2 I1=0
where In = 0 implies a perfect open-circuit at port n. These definitions can be writ-ten as
3V14= 3Z11 Z1243I14 (2.20)
The matrix, which contains the Z parameters, is known as the open-circuit imped-ance or Z matrix and is denoted by [Z]. For reciprocal networks, Z12 = Z21. If net-works are symmetrical, Z12 = Z21 and Z11 = Z22. For a lossless network, the Z para-meters are all purely imaginary.
Inspecting (2.18) and (2.20), we immediately obtain an important relation
[Z] = [Y]–1 (2.21)
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